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Chapter 9. Linear Momentum and Collisions. Intro. Consider bowling: Bowling ball collides with initial pin Force on/Acceleration of the Pin Force on/Acceleration of the ball Momentum- simplified way to study these moving objects. 9.1 Linear Momentum and Conservation.

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chapter 9

Chapter 9

Linear Momentum and Collisions

intro
Intro
  • Consider bowling:
    • Bowling ball collides with initial pin
      • Force on/Acceleration of the Pin
      • Force on/Acceleration of the ball
  • Momentum- simplified way to study these moving objects.
9 1 linear momentum and conservation
9.1 Linear Momentum and Conservation

Consider two particles (isolated) m1 and m2 moving at v1 and v2

From Newton’s 3rd Law

slide4
9.1
  • And if the the masses are constant
  • If the derivative of a function is 0 it is constant (conserved)
slide5
9.1
  • Linear Momentum- the product of the mass and velocity of a moving particle.
    • Momentum is a vector quantity
    • Has dimensions MLT-1 and SI units kg.m/s
    • All momentum is conserved
    • Three Components of Momentum
slide6
9.1
  • Directly related to Newton’s 2nd Law
      • Instead of Net Force equals mass times accel.
      • Can be described as Net Force equals time rate of change of momentum
slide7
9.1
  • Conservation of Momentum
    • For isolated systems the time derivative of the total momentum is 0
    • The total momentum is therefore constant or conserved.

Law of Conservation of Momentum

    • And in 3 components
slide8
9.1
  • Quick Quizzes p 254-255
  • Examples 9.1-9.2
9 2 impulse and momentum
9.2 Impulse and Momentum
  • The momentum of an object changes when a net force acts on it

or

  • Integrating this gives
slide10
9.2
  • Impulse-Momentum Theorem- the impulse of the force F acting on a particle equals the change in momentum of the particle
  • Force Varies, impulse time is short impulse can generally be calculated with the average force.
slide11
9.2
  • Quick Quizzes p. 258
  • Examples 9.3-9.4
9 3 collisions in 1 d
9.3 Collisions in 1-D
  • Momentum is conserved
  • Three types of Collisions
    • Inelastic
    • Perfectly Inelastic
    • Elastic
slide13
9.3
  • Inelastic Collision- Momentum is conserved but kinetic energy is not.
    • Inelastic- objects collide and separate, some K is lost
    • Perfectly Inelastic- objects collide and stick together (moving as one), some K is lost
slide14
9.3
  • Elastic Collisions-A collision in which no energy is lost to (surroundings / internal / potential)
    • Both momentum and kinetic energy are conserved
slide15
9.3
  • By combining the Conservation of p and K equations
  • In all collision types careful attention to the direction (and sign) of velocities must be paid.
slide16
9.3
  • Quick Quizzes p. 262
  • Examples 9.5 - 9.9
9 4 2 d collisions
9.4 2-D Collisions
  • Momentum is conserved on each axis
  • Examples 9.10 – 9.12
9 5 center of mass
9.5 Center of Mass
  • We can describe the overall motion of a mechanical system by tracking its center of mass
    • System could be a group of particles
    • System could be a large extended object
  • A force applied to the center of mass will cause no rotation to the system
slide19
9.5
  • To find the center of mass in 3-D space for a number (i) particles
  • Or in terms of the position vector of each particle
slide20
9.5
  • For extended objects that have a continuous mass distribution
  • Consider them an infinite number of closely spaced particles
  • The sum becomes an integral
slide21
9.5
  • Or in terms of the position vector
  • For symmetrical objects, the center of mass lies on the axis/plane of symmetry
    • Examples: uniform rod,

sphere,

cube,

donut?

slide22
9.5
  • For extended objects, the force of gravity acts individually on each small piece of mass (dm)
  • The net effect of all these forces is equivalent to the single force Mg, through a point called the center of gravity.
  • If the gravitational field is uniform across all dm, the center of gravity and center of mass are one and the same.
slide23
9.5
  • Quick quiz p 272
  • Examples 9.13, 9.14, 9.15
9 6 motion of a system
9.6 Motion of a System
  • If the mass of a system remains constant (no particles entering/leaving) then we can track the motion of the center of mass, rather than the individual particles.
  • Also assumes any forces on the system are internal (isolated)
slide25
9.6
  • Velocity of the center of mass
  • Acceleration of the center of mass
slide26
9.6
  • If there is a net force on the system, it will move equivalent to the way a single M with the same net force would move.
  • And if the net force is zero
slide27
9.6
  • Quick Quizzes p. 276
  • Examples 9.17, 9.18
9 7 rocket propulsion
9.7 Rocket Propulsion
  • Most forms of vehicular motion result from action/reaction friction.
  • A rocket has nothing to push against so its motion/control depend on conservation of motion of the system.
  • The system includes the rocket body (and payload) plus the ejected fuel
slide29
9.7
  • The rocket burns fuel and oxidizer creating expanding gases that are directed through the nozzle.
  • Each gas molecule has a mass (that was once part of the rockets total mass) and velocity, therefore a downward momentum.
  • The rocket receives the same compensating momentum upward.
slide30
9.7
  • Looking a rocket initially with mass M + Δm, moving with velocity v…
slide31
9.7
  • And some time, Δt, later...
  • The rocket now has mass, M and velocity v + Δv, compensating the momentum of the exhausted mass, Δm.
slide32
9.7
  • The conservation of momentum expression for this change…
  • Can be simplified to…
slide33
9.7
  • A rocket motor produces a continuous flow of exhaust gas a fairly constant speed, through the burn
  • For continually changing values…

Δv dv

Δm dm

So… 

slide34
9.7
  • Because the increase in exhaust mass = the decrease in rocket mass…
  • Then integrate this expression
slide35
9.7
  • Discuss integral of M-1
  • Evaluating from vi to vf gives the basic expression for rocket propulsion.
slide36
9.7
  • Mi is the total mass of the rocket/payload plus fuel
  • Mf is the mass of the rocket/payload
  • Mi – Mf is the mass of fuel needed to achieve a certain speed (eg. Escape speed to power down rocket)
slide37
9.7
  • Thrust- the actual force on the rocket at any given time is
  • Thrust is proportional to exhaust speed and also the rate of change of mass (burn rate).
  • Examples 9.19 p. 279
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